There are two items on the Options screen that deal with this exact issue. To be able to see completed activities and projects, make sure that these items are both selected.
"Show completed projects"
"Show completed activities"
These options are user-specific, not global. That means that if one user (even an administrator) changes them, the change will not affect what other users see.
Note that a quicker way to enable showing completed objects is to use links in the Quick Panel:
Article ID: 15, Created On: 2/26/2008, Modified: 3/5/2012